What would happen if a huge box were built (as tall as 40 football fields) with scales under it, measuring its weight with birds inside? When the birds take flight they push air downwards, keeping the weight essentially the same… But if the birds gAin more height… Would the effect still be present? I mean, we are not crushed by air if a helicopter flies over our heads…
If it’s a closed box, the movement of objects inside it may cause momentary and self-cancelling fluctuations on your scale, but the weight of the system would not change.
An easier way to think of this is a fishtank. It weighs the same regardless whether the fish are swimming about, or resting on the bottom.
The force is spread out.
There is noting special about flying, although our intuition seems to say that it is. Flying in a closed box is identical to climbing a rope in a closed box. So try the thought experiment with a flock of monkeys (so what is the collective noun here?) and a lot of ropes hanging from the top of the box. What happens? In a steady state the box weights the same, if the monkeys are moving steadily, still the same. If the monkeys are starting or stopping, or accelerating in any way, the weight of box as measured by a set of scales outside changes in response to the force changes the monkeys provide. In one limit, if all the monkeys climbed to the top of the box, and simultaneously jumped into the air, for the time it takes them to reach the bottom of the box, the box would weigh less by the weight of the monkeys. That is no different to all the birds in a box flying into a dive. When the monkeys hit the floor, or the birds pull out of their dive, there will be a transient acceleration that makes the box weigh more.
After that is simply a matter of whether you have European or African swallows, or the box is on a treadmill.
Yes, it’s spread out over the area of a circle equal to the main rotor diameter. I’ve witnessed a demonstration in which an RC model helicopter is parked on a sheet of cardboard, which in turn is resting on a scale. The scale registers the actual weight of the helicopter. The operator then fires up the helicopter and puts it in a hover a couple of inches above the cardboard. The rotor downwash striking the cardboard results in a downforce that is measured by the scale - and just happens to exactly equal the weight of the helicopter.
Troop. Or tribe. Or barrel.
You ask, SD delivers.
African or European?
This is not the same. The helicopter is not in a closed space, and is only a “couple of inches” from the cardboard.
In the OP the birds are in a “huge box” resting on a scale, would the downforce of their beating wings at a much greater distance (40 football fields) from the bottom still effect the scale?
If the walls of the box are all enclosed, then yes. If they’re not, then the downward-moving air can move out of the box to some degree, and the box will be lighter.
Yes - if it’s a closed box, how could it not affect it?
The system of you + a ladder weighs the same regardless whether you are on the top or bottom rung (ignoring silly things like the tiny change in the force of gravity as you climb)
Again, not the same, the ladder must be resting on the bottom of the box.
Absolutely the same. Air is just a little less substantial than the metal or wood of your ladder.
A bird flying in a vacuum inside a sealed box would not contribute to the weight of the box.
But birds can’t fly (let alone live) in a vacuum.
If some downward-moving air isn’t intercepted by the bottom of the box, then the only other thing that can arrest its movement is the ground - upon which it will exert a downward force equal to the difference between the birds’ weight and the force exerted on the bottom of the box by the rest of the downward-moving air.
This. Want to stay above the floor?
You can push down on the floor directly with your bare feet.
Or you
You can push down on your shoes, which in turn push down on the floor.
Or
You can push down on a ladder, which in turn pushes down on the floor.
Or
You can push down on the air, which in turn pushes down on the floor.
OK, get it now. I was thinking the distance from the scale would cause less downward force and be spread out over more area. The scale only measuring the downforce and not lateral.
Thanks for clearing that up, I may need a nap now. 
What happens if there’s a treadmill in the box and the bird is trying to take off?
I think the answer to that is plane.
Same result as theMythbusters got. The average weight of the box will remain the same as the birds fly. Giving yourself a bigger box just makes any effects harder to measure.
Does the volume of the box and height of the helicopter change anything?
For instance using a 10’ square sealed box sitting on a scale. Using a miniature .5 pound RC helicopter and hovering at the 9’ mark does the scale still read .5?